3 Answers
3

To show continuity at $x=0$, you're being asked to show
$$
\lim_{x\to0} |x|\sin(\cot x) = \text{the given value} = 0.
$$
Since $\sin(\text{anything})$ is between $1$ and $-1$ (inclusive), you've got $|x|\sin(\text{something})$ between $|x|$ and $-|x|$, and those both approach $0$ as $x\to 0$, so that's all you need to do for that part.

For the other part, you need to show that
$$
\lim_{x\to1/42} |x|\sin(\cot x) \ne \text{the given value} = 10^{42}.
$$
Here it would suffice to show that

$|1/42|\sin(\cot(1/42))$ is negative and

since the function is continuous, it must remain negative in some neighborhood of $x=1/42$ and